A Partially Implicit Method for Large Stiff Systems of ODE<scp>s</scp> with Only Few Equations Introducing Small Time-Constants

Hofer, Eberhard P.

doi:10.1137/0713054

Cited by 70 publications

(41 citation statements)

References 17 publications

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“…It seems inefficient to solve these problems only with implicit methods. If the problem can be partitioned into a stiff part and a nonstiff part such as (8) where are representations of stiff and nonstiff subsystems with variables and . respectively.…”

Section: Decoupled Methodsmentioning

confidence: 99%

“…In the mathematical literature, the idea of partitioning a stiff system into a small stiff system and a large nonstiff part goes back to Hofer [8]. In [8], the differential equations are decomposed by the time constants of the equations where a few distinct equations introducing the small time constants are considered the stiff part.…”

Section: Introductionmentioning

confidence: 99%

“…In [8], the differential equations are decomposed by the time constants of the equations where a few distinct equations introducing the small time constants are considered the stiff part. As stiffness may affect subspaces that are not parallel to the coordinate axis, the projection methods are applied as generalized partition methods later.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Decoupled Time-Domain Simulation Method via Invariant Subspace Partition for Power System Analysis

Yang

Ajjarapu

2006

IEEE Trans. Power Syst.

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Abstract-A decoupled method is proposed to deal with timedomain simulation for power system dynamic analysis. Traditionally, there are two main categories of numerical integration methods: explicit methods and implicit methods. The implicit methods are numerically stable but require more computational time to solve the nonlinear equations, while explicit methods are relatively efficient but may cause a numerical stability problem. This paper proposes a new hybrid method to take advantage of both explicit and implicit methods based on the invariant subspace partition. The original power system equations are decoupled into two parts that correspond to the stiff and nonstiff subspaces. For the stiff invariant subspace, the implicit method is applied to achieve numerical stability, and the explicit method is employed to handle nonstiff invariant subspace for the computational efficiency. As a result, the new hybrid method is both numerically stable and efficient. The approach is demonstrated through New England 39-bus and IEEE 118-bus systems.Index Terms-A-stability, decoupled method, differential and algebraic equations (DAE), dynamical simulation, explicit methods, implicit methods, invariant subspace partition, power systems, stiff systems, time-domain simulation.

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Section: Decoupled Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Decoupled Time-Domain Simulation Method via Invariant Subspace Partition for Power System Analysis

Yang

Ajjarapu

2006

IEEE Trans. Power Syst.

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“…Here, step-sizes of integrators for different subsets of variables are the same, but different types of integrators are used for different subsets. Examples of this approach include the work of Gunther and Rentrop (1994), Hofer (1976), Rentrop (1985), Strehmel and Weiner (1984), and Weiner et al (1993). Another variation of this approach is discussed by Enright and Kamel (1979) and by Gear and Saad (1983).…”

Section: Introductionmentioning

confidence: 99%

Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems

Shome

Haug

Jay

2004

Mechanics Based Design of Structures and Machines

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A framework is presented allowing dual-rate numerical integration of the equations of mechanical system dynamics to be considered as a form of Partitioned Runge-Kutta (PRK) integration. Certain coefficients of a PRK integrator are set to zero, so that Runge-Kutta integrators that constitute the PRK integrator can be made to have different numbers of stages. As a result, one Runge-Kutta integrator requires fewer function evaluations than the other does, which is a form of dual-rate integration. Well-established order of accuracy theory for PRK integrators is used to develop a rigorous methodology for designing explicit PRK dual-rate integrators. Stabilized Runge-Kutta theory developed for single-rate RungeKutta integrators is combined with PRK integrator theory to design PRK dual-rate integrators with the largest possible stability regions. Dual-rate PRK integrators created using these approaches are used to simulate the dynamics of vehicle systems that contain subsystems with higher frequency response characteristics than do the basic vehicle subsystems.

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“…First results for Runge-Kutta [13] and Rosenbrock-Wanner schemes [14] have been generalized to linearly-implicit RK methods [24]. Applications: chemical reaction kinetics and circuit simulation.…”

mentioning

confidence: 99%

Minisymposium Multirate Time Integration for Multiscaled Systems

Maten

Günther

2010

Progress in Industrial Mathematics at ECMI 2008

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A Partially Implicit Method for Large Stiff Systems of ODEs with Only Few Equations Introducing Small Time-Constants

Cited by 70 publications

References 17 publications

A Decoupled Time-Domain Simulation Method via Invariant Subspace Partition for Power System Analysis

A Decoupled Time-Domain Simulation Method via Invariant Subspace Partition for Power System Analysis

Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems

Minisymposium Multirate Time Integration for Multiscaled Systems

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